MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

138 Mathematical Competitions
Problem 30 (IMO Supplemental). Two equilateral triangles are inscribed
in a circle with radius r. Let K be the area of the set consisting of all points
interior to both triangles. Prove that K ≥ r 2√ 3/2. [195, p. 13]
Problem 31 (IMO 1986). Given a triangle A1A2A3 and a point P0 in
the plane, define As = As−3 for all s ≥ 4. Construct a sequence of points
P1, P2, P3, . . . such that Pk+1 is the image of Pk under rotation with center Ak+1
through angle 120 ◦ clockwise (for k = 0, 1, 2, . . .). Prove that if P1986 = P0 then
the triangle A1A2A3 is equilateral. [200, p. 1]
Figure 5.15: IMO 2005
Problem 32 (IMO 2005). Six points are chosen on the sides of an equilateral
triangle ABC: A1 and A2 on BC, B1 and B2 on CA, and C1 and C2 on AB
(Figure 5.15). These points are the vertices of a convex equilateral hexagon
A1A2B1B2C1C2. Prove that lines A1B2, B1C2, and C1A2 are concurrent (at
the center of the triangle). [103, p. 5]
Problem 33 (Austrian-Polish Mathematics Competition 1989). If
each point of the plane is colored either red or blue, prove that some equilateral
triangle has all its vertices the same color. [182, p. 42]